Introductory Notes on Valuation Rings and Function Fields in One Variable

  • Renata Scognamillo
  • Umberto Zannier

Part of the Publications of the Scuola Normale Superiore book series (PSNS, volume 14)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Renata Scognamillo, Umberto Zannier
    Pages 1-6
  3. Renata Scognamillo, Umberto Zannier
    Pages 7-29
  4. Renata Scognamillo, Umberto Zannier
    Pages 31-84
  5. Back Matter
    Pages 85-119

About this book


The book deals with the (elementary and introductory) theory of valuation rings. As explained in the introduction, this represents a useful and important viewpoint in algebraic geometry, especially concerning the theory of algebraic curves and their function fields. The correspondences of this with other viewpoints (e.g. of geometrical or topological nature) are often indicated, also to provide motivations and intuition for many results. Links with arithmetic are also often indicated. There are three appendices, concerning Hilbert’s Nullstellensatz (for which several proofs are provided), Puiseux series and Dedekind domains. There are also several exercises, often accompanied by hints, which sometimes develop further results not included in full for brevity reasons.


Dedekind domains Hilbert's Nullstellensatz Puiseux series valuation rings

Authors and affiliations

  • Renata Scognamillo
    • 1
  • Umberto Zannier
    • 1
  1. 1.Scuola Normale SuperiorePisaItaly

Bibliographic information